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Meeting 33

Thursday, 25 January 2024

Hosted by the University of Leeds
Supported by the London Mathematical Society

Schedule:

  • 13:00 -- 14:00 Mike Prest (University of Manchester).
  • 14:00 -- 14:30 Coffee break.
  • 14:30 -- 15:30 Ana Ros Camacho (University of Cardiff).
  • 15:30 -- 16:00 Coffee break.
  • 16:00 -- 17:00 Giacomo Tendas (Masaryk University / University of Manchester).

Location:

The meeting will be held in the School of Mathematics at the University of Leeds (A link to the campus map is here).

Registration:

The registration is free, but please register using this form. Registration closes on 22nd January 2024 at 5 pm (UK time).

Titles and Abstracts:

Speaker: Mike Prest
Title: Definable categories

Abstract: Definable additive categories arose from the model theory of modules; they give the "right" context in which to do model theory of additive structures. To each definable category there is associated a small abelian category and this leads to an anti-equivalence between the 2-category of definable additive categories with definable functors and the 2-category of skeletally small abelian categories with exact functors.

 

Speaker: Ana Ros Camacho
Title: Reconstructing algebra objects from NIM-reps in pointed, near-group and quantum group-like fusion categories

Abstract: Algebras in tensor categories are interesting mathematical objects, related to conformal and topological field theory. However, they are a bit tricky to find and describe explicitly. In this work, we compute Morita equivalence classes of algebras coming from the Non-negative Integer Matrix representations (or simply NIM-reps) of the fusion rings of pointed, near-group and A(1,k)_{1/2}, and compare these results with some existing ones. Joint work with Sam Hannah

 

Speaker: Giacomo Tendas
Title: Absolute colimits in enriched category theory

Abstract: A colimit of a certain shape is called "absolute" if it is preserved by any functor. A question that naturally arises is: Can we characterize absolute colimits in a meaningful and explicit way? The answer, in ordinary category theory, is well known and quite easy to express: absolute colimits are generated by coequalizers of split pairs (or equivalently, by splittings of idempotents).
When moving to the context of enriched category theory, however, the situation becomes quite tricky, as one now works with the more general notion of "weighted colimit". For instance, enriching over the poset of positive real numbers (so that V-categories are Lawvere metric spaces) limits of Cauchy sequences are an example of enriched absolute colimits. This witnesses how much the answer to the question above depends on the base of enrichment, making a general approach to the problem almost impossible.
In this talk I will explain why absolute colimits are important and give a variety of examples of bases of enrichment for which it is possible to answer the question in a satisfactory manner; these include many bases of interest, e.g: Ab, R-Mod, DGAb, Cat, SSet, Met, etc.